It's a two-year, $19-million contract, Haynes reports.
Thompson has spent his entire career with the Cleveland Cavaliers, who drafted him with the fourth overall pick in 2011.
The Cavaliers tried to convince Thompson to return, but their offer was for less than what the Celtics were willing to pay, according to The Athletic's Joe Vardon.
Thompson averaged 12 points and 10.1 rebounds in 57 contests this past season. He ranks seventh on the Cavaliers' all-time games played list (619), third in rebounds (5,393), and sixth in blocks (447).